I got the original boxed game from WAY back in time, some 13-14 years or so. I reinstalled the game after having it shelved for a long time and installed the newest patch. Will Alexandria work if I buy that from steam of will I have to buy the whole game from steam for it to work?

As I recall it, the enhanced edition was effectively a free patch that came out when TM bought the rights back to their own game (from Sega in Europe and whoever in USA). It fixed a few bugs, added the brickyard and removed the sun and moon. Saved games went in a Titled Mill subdirectory of Documents instead of Sega. Probably there were a few things I've forgotten about, but there should be the original posts about it on the forum if you search for it

Alexandria you had to pay extra for.

Note that you can also get CotN complete from GOG.com (presumably this includes Alexandria) if that is cheaper or more convenient.

Alexandria will install with the original boxed version of the game. It actually forces a download of the streaming version from STEAM and creates a new set of folders for the streaming version in which to install. The boxed version files remain on the hard drive and can be accessed and played via the game disc as normal too.